Defects and polarity. Assuming 2D periodic boundary conditions
and using a Fourier pseudo-spectral numerical method
[1]}, [2]}, we simulated our model over long times. In
regions of flow instability, we find persistently unsteady
turbulent-like flows that are correlated with continual genesis,
propagation, and annihilation of \(\pm 1/2\)  order defect pairs (see
videos S2–S5). In Fig. REF a, the defects exist in regions
of small nematic order (dark blue), and are born as opposing pairs in
elongated “incipient crack” regions, qualitatively similar to the
structures found in both experiments and apolar models
[3]}, [4]}, [5]}, [6]}, [7]}. Moreover, as shown
in panel (b), the polarity field develops considerable spatial
variation with regions of high and low polar order \(|{\bf q}|\)  (video S5).
The two active stresses vary in strength depending on the local
polarity — the polar-aligned (anti-aligned) stress is large in
regions of high (low) polar order (panels (c) & (d)) — and hence
are largest respectively in their complementary regions. The circles
in Fig. REF b encircle \(+1/2\) -order defects, showing the
sharp variation of the polarity field around them, with the gradients
of active stresses there yielding large active force shown in panel
(e).
